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Big Data and Cognitive Computing

Deep Learning-Based Method for a Ground-State Solution of Bose-Fermi Mixture at Zero Temperature

Xianghong He, Jidong Gao, Rentao Wu, Yuhan Wang, Rongpei Zhang

Published Nov 4, 2025• 0 citations
Publisher

Abstract

A Bose-Fermi mixture, consisting of both bosons and fermions, exhibits distinctive quantum coherence and phase transitions, offering valuable insights into many-body quantum systems. The ground state, as the system’s lowest energy configuration, is essential for understanding its overall behavior. In this study, we introduce the Bose-Fermi Energy-based Deep Neural Network (BF-EnDNN), a novel deep learning approach designed to solve the ground-state problem of Bose-Fermi mixtures at zero temperature through energy minimization. This method incorporates three key innovations: point sampling pre-training, a Dynamic Symmetry Layer (DSL), and a Positivity Preserving Layer (PPL). These features significantly improve the network’s accuracy and stability in quantum calculations. Our numerical results show that BF-EnDNN achieves accuracy comparable to traditional finite difference methods, with effective extension to two-dimensional systems. The method demonstrates high precision across various parameters, making it a promising tool for investigating complex quantum systems.

DOI
10.3390/bdcc9110279

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A Bose-Fermi mixture, consisting of both bosons and fermions, exhibits distinctive quantum coherence and phase transitions, offering valuable insights into many-body quantum systems. The ground state, as the system’s lowest energy configuration, is essential for understanding its overall behavior. In this study, we introduce the Bose-Fermi Energy-based Deep Neural Network (BF-EnDNN), a novel deep learning approach designed to solve the ground-state problem of Bose-Fermi mixtures at zero temperature through energy minimization. This method incorporates three key innovations: point sampling pre-training, a Dynamic Symmetry Layer (DSL), and a Positivity Preserving Layer (PPL). These features significantly improve the network’s accuracy and stability in quantum calculations. Our numerical results show that BF-EnDNN achieves accuracy comparable to traditional finite difference methods, with effective extension to two-dimensional systems. The method demonstrates high precision across various parameters, making it a promising tool for investigating complex quantum systems.
Abstract This paper introduces a novel numerical method based on an energy-minimizing normalized residual network (EM-NormResNet) to compute the ground-state solution of Bose–Einstein condensates at zero or low temperatures. Starting from the three-dimensional Gross–Pitaevskii equation (GPE), we reduce it to the 1D and 2D GPEs because of the radial symmetry and cylindrical symmetry. The ground-state solution is formulated by minimizing the energy functional under constraints, which is directly solved using the EM-NormResNet approach. The paper provides detailed solutions for the ground states in 1D, 2D (with radial symmetry), and 3D (with cylindrical symmetry). We use the Thomas–Fermi approximation as the target function to pre-train the neural network. Then, the formal network is trained using the energy minimization method. In contrast to traditional numerical methods, our neural network approach introduces two key innovations: (i) a novel normalization technique designed for high-dimensional systems within an energy-based loss function; (ii) improved training efficiency and model robustness by incorporating gradient stabilization techniques into residual networks. Extensive numerical experiments validate the method’s accuracy across different spatial dimensions.
Previous numerical studies on the ground state structure of Bose--Fermi mixtures mostly relied on Thomas--Fermi (TF) approximation for the Fermi gas. In this paper, we establish the existence and uniqueness of ground state solutions of Bose--Fermi mixtures at zero temperature for both a coupled Gross--Pitaevskii (GP) equations model and a model with TF approximation for fermions. To prove the uniqueness, the key is to estimate the L∞L^\inftyL∞ bounds of the ground state solution. By implementing an efficient method---gradient flow with discrete normalization with backward Euler finite difference discretization---to compute the coupled GP equations, we report extensive numerical results in one and two dimensions. The numerical experiments show that we can also extract many interesting phenomena without reference to TF approximation for the fermions. Finally, we numerically compare the ground state solutions for the coupled GP equations model and the model with TF approximation for fermions as well as for the model with TF approximations for both bosons and fermions.
The simulation of quantum many-body systems poses a significant challenge in physics due to the exponential scaling of Hilbert space with the number of particles. Traditional methods often struggle with large system sizes and frustrated lattices. In this research article, we present a novel algorithm that leverages the power of deep neural networks combined with Markov Chain Monte Carlo simulation to address these limitations. Our method introduces a neural network architecture specifically designed to represent bosonic quantum states on a 1D lattice chain. We successfully achieve the ground state of the Bose-Hubbard model, demonstrating the superiority of the adaptive momentum optimizer for convergence speed and stability. Notably, our approach offers flexibility in simulating various lattice geometries and potentially larger system sizes, making it a valuable tool for exploring complex quantum phenomena. This work represents a substantial advancement in the field of quantum simulation, opening new possibilities for investigating previously challenging systems.
Deep learning techniques have opened a new venue for electronic structure theory in recent years. In contrast to traditional methods, deep neural networks provide much more expressive and flexible wave function Ans\"atze, resulting in better accuracy and timescale behavior. In order to study larger systems while retaining sufficient accuracy, we integrate a powerful neural-network-based model (FermiNet) with the effective core potential method, which helps to reduce the complexity of the problem by replacing inner core electrons with additional semilocal potential terms in the Hamiltonian. In this work, we calculate the ground-state energy of 3d3d3d transition metal atoms and their monoxides, which is quite challenging for the original FermiNet work, and the results are consistent with both experimental data and other state-of-the-art computational methods. Our work is an important step for a broader application of deep learning in the electronic structure calculation of molecules and materials.
Abstract Machine learning has rapidly become a powerful tool for addressing challenges in ultracold atomic systems; however, its application to intricate three-dimensional (3D) systems remains relatively underexplored. In this study, we introduce a 3D residual network (3D ResNet) framework based on 3D convolutional neural networks (3D CNN) to predict ground states phases in 3D dipolar spinor Bose–Einstein condensates (BECs). Our results show that the 3D ResNet framework predicts ground states with high accuracy and efficiency across a broad parameter space. To enhance phase transition predictions, we incorporate data augmentation techniques, leading to a notable improvement in the model’s performance. The method is further validated in more complex scenarios, particularly when transverse magnetic fields are introduced. Compared to conventional imaginary-time evolution methods (ITEM), the 3D ResNet drastically reduces computational costs, offering a rapid and scalable solution for complex 3D multi-parameter nonlinear systems.
Quantum theory has been remarkably successful in providing an understanding of physical systems at foundational scales. Solving the Schrödinger equation provides full knowledge of all dynamical quantities of the physical system. However closed form solutions to this equation are only available for a few systems and approximation methods are typically used to find solutions. In this paper we address the problem of solving the time independent Schrödinger equation for the ground state solution of physical systems. We propose using end-to-end deep learning approach in a variational optimization scheme for approximating the ground state energies and wave functions of these systems. A neural network realizes a universal trial wave function and is trained in an unsupervised learning framework by optimizing the expectation value of the Hamiltonian of a physical system. The proposed approach is evaluated on physical systems consisting of a particle in a box with and without a perturbation. We demonstrate that our approach obtains approximations of ground state energies and wave functions that are highly accurate, which makes it a potentially plausible candidate for solving more complex physical systems for which analytical solutions are beyond reach.